Coexistent phase mixtures in the anti-plane shear of an elastic tube

We study the detailed structure of the deformed configuration of an elastic tube whose cross section is a convex ring that is subjected to a prescribed relative axial displacement of its lateral boundaries. The material is assumed to have a non-convex stored energy function. Special attention is paid to the situation when there is no minimizer of the associated anti-plane shear minimization problem, but, nevertheless, the energy functional has an infimum. The non-existence of a minimizer to this problem for a certain interval of prescribed relative axial displacement of the lateral boundaries implies that among all "admissible" deformations there is none with this boundary data for which the values of the stored energy function correspond to its convex points almost everywhere in the body. Because of this, we find that to reach the infimum the tube divides into three subdomains: one of high strain, one of low strain, and one of intermediate "mixed" strain. In the intermediate "mixed" strain subdomain, the field values of the stored energy correspond to convex combinations of convex, but not strictly convex, points of the stored energy function. The main variational problem then gives rise to a free boundary problem in which the subdomain where the strict convexity of the stored energy function breaks down must be determined as part of the solution. The characterization of this intermediate phase mixture region is one of the goals of this work.